Scholarly Works - Pure Mathematics and Mathematical Statistics
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Item Open Access Published version Peer-reviewed Efficient nonparametric bayesian inference for X-ray transforms(Institute of Mathematical Statistics, 2019) Monard, F; Nickl, R; Paternain, GPWe consider the statistical inverse problem of recovering a function $f: M \to \mathbb R$, where $M$ is a smooth compact Riemannian manifold with boundary, from measurements of general $X$-ray transforms $I_a(f)$ of $f$, corrupted by additive Gaussian noise. For $M$ equal to the unit disk with `flat' geometry and $a=0$ this reduces to the standard Radon transform, but our general setting allows for anisotropic media $M$ and can further model local `attenuation' effects -- both highly relevant in practical imaging problems such as SPECT tomography. We propose a nonparametric Bayesian inference approach based on standard Gaussian process priors for $f$. The posterior reconstruction of $f$ corresponds to a Tikhonov regulariser with a reproducing kernel Hilbert space norm penalty that does not require the calculation of the singular value decomposition of the forward operator $I_a$. We prove Bernstein-von Mises theorems that entail that posterior-based inferences such as credible sets are valid and optimal from a frequentist point of view for a large family of semi-parametric aspects of $f$. In particular we derive the asymptotic distribution of smooth linear functionals of the Tikhonov regulariser, which is shown to attain the semi-parametric Cram\'er-Rao information bound. The proofs rely on an invertibility result for the `Fisher information' operator $I_a^*I_a$ between suitable function spaces, a result of independent interest that relies on techniques from microlocal analysis. We illustrate the performance of the proposed method via simulations in various settings.Item Open Access Accepted version Peer-reviewed SENSITIVITY OF MIXING TIMES IN EULERIAN DIGRAPHS(Society for Industrial & Applied Mathematics (SIAM), 2018) Boczkowski, Lucas; Peres, Yuval; Sousi, PerlaItem Open Access Published version Peer-reviewed On the p ′ -Subgraph of the Young Graph(Springer Science and Business Media LLC, 2019) Giannelli, E; Law, S; Martin, S; Law, Stacey [0000-0001-7936-0938]Let $p$ be a prime number. In this article we study the restriction to $\mathfrak{S}_{n-1}$ of irreducible characters of degree coprime to $p$ of $\mathfrak{S}_n$. In particular, we study the combinatorial properties of the subgraph $\mathbb{Y}_{p'}$ of the Young graph $\mathbb{Y}$. This is an extension to odd primes of the work done by Ayyer, Prasad and Spallone for $p=2$.Item Open Access Published version Peer-reviewed Coupling the Gaussian Free Fields with Free and with Zero Boundary Conditions via Common Level Lines(Springer Nature, 2018-07-01) Qian, W; Werner, Wendelin; Qian, Wei [0000-0002-4779-4042]We point out a new simple way to couple the Gaussian free field (GFF) with free boundary conditions in a two-dimensional domain with the GFF with zero boundary conditions in the same domain: Starting from the latter, one just has to sample at random all the signs of the height gaps on its boundary-touching zero level lines (these signs are alternating for the zero-boundary GFF) in order to obtain a free boundary GFF. Constructions and couplings of the free boundary GFF and its level lines via soups of reflected Brownian loops and their clusters are also discussed. Such considerations show for instance that in a domain with an axis of symmetry, if one looks at the overlay of a single usual Conformal Loop Ensemble CLE3 with its own symmetric image, one obtains the CLE4-type collection of level lines of a GFF with mixed zero/free boundary conditions in the half-domain.Item Open Access Published version Peer-reviewed Free by cyclic groups and linear groups with restricted unipotent elements(Walter de Gruyter GmbH, 2017-01-24) Button, JOAbstractWe introduce the class of linear groups that do not contain unipotent elements of infinite order, which includes all linear groups in positive characteristic. We show that groups in this class have good closure properties, in addition to having properties akin to non-positive curvature, which were proved in [Item Open Access Accepted version Peer-reviewed Computing JSJ decompositions of hyperbolic groups(OUP) Barrett, BJWe present an algorithm that computes Bowditch's canonical JSJ decomposition of a given one-ended hyperbolic group over its virtually cyclic subgroups. The algorithm works by identifying topological features in the boundary of the group. As a corollary we also show how to compute the JSJ decomposition of such a group over its virtually cyclic subgroups with infinite centre. We also give a new algorithm that determines whether or not a given one-ended hyperbolic group is virtually fuchsian. Our approach uses only the geometry of large balls in the Cayley graph and avoids Makanin's algorithm.Item Open Access Accepted version Peer-reviewed Impact of Cultural Exposure and Message Framing on Oral Health Behavior: Exploring the Role of Message Memory.(SAGE Publications, 2016-10) Brick, Cameron; McCully, Scout N; Updegraff, John A; Ehret, Phillip J; Areguin, Maira A; Sherman, David K; Brick, Cameron [0000-0002-7174-8193]BACKGROUND: Health messages are more effective when framed to be congruent with recipient characteristics, and health practitioners can strategically choose message features to promote adherence to recommended behaviors. We present exposure to US culture as a moderator of the impact of gain-frame versus loss-frame messages. Since US culture emphasizes individualism and approach orientation, greater cultural exposure was expected to predict improved patient choices and memory for gain-framed messages, whereas individuals with less exposure to US culture would show these advantages for loss-framed messages. METHODS: 223 participants viewed a written oral health message in 1 of 3 randomized conditions-gain-frame, loss-frame, or no-message control-and were given 10 flosses. Cultural exposure was measured with the proportions of life spent and parents born in the US. At baseline and 1 week later, participants completed recall tests and reported recent flossing behavior. RESULTS: Message frame and cultural exposure interacted to predict improved patient decisions (increased flossing) and memory maintenance for the health message over 1 week; for example, those with low cultural exposure who saw a loss-frame message flossed more. Incongruent messages led to the same flossing rates as no message. Memory retention did not explain the effect of message congruency on flossing. LIMITATIONS: Flossing behavior was self-reported. Cultural exposure may only have practical application in either highly individualistic or collectivistic countries. CONCLUSIONS: In health care settings where patients are urged to follow a behavior, asking basic demographic questions could allow medical practitioners to intentionally communicate in terms of gains or losses to improve patient decision making and treatment adherence.Item Open Access Accepted version Peer-reviewed When stereotype threat does not impair performance, self-affirmation can be harmful(Informa UK Limited, 2019) Voisin, D; Brick, C; Vallée, B; Pascual, A; Brick, Cameron [0000-0002-7174-8193]Item Open Access Published version Peer-reviewed Personality trait effects on green household installations(University of California Press, 2018) Busic-Sontic, A; Brick, C; Brick, Cameron [0000-0002-7174-8193]Large, one-time investments in green energy installations effectively reduce domestic energy use and greenhouse gas emissions. Despite long-term economic benefits for households, the rate of green investments often remains moderate unless supported by financial subsidies. Beyond financial considerations, green investments may also be driven by individual psychological factors. The current study uses data from the German Socio-Economic Panel (N = 3,468) to analyse whether the household decision to invest in green energy installations is linked to the Big Five personality traits. Personality traits and domestic investments in solar and other alternative energy systems had weak indirect associations through environmental concern but not through risk preferences. Openness to Experience and Neuroticism showed a weak positive relationship with green energy installations through the environmental concern channel, whereas Extraversion had a weak negative link. Based on these findings, persuasive messaging for green investments may be more effective when it focuses on environmental concern rather than reduced risk in countries like Germany, where long-standing financial subsidies decreased the risk in green investments.Item Open Access Accepted version Peer-reviewed Nearby Lagrangian fibers and Whitney sphere links(Wiley, 2018) Ekholm, Tobias; Smith, IvanLet $n>3$, and let $L$ be a Lagrangian embedding of $\mathbb{R}^{n}$ into the cotangent bundle $T^{\ast }\mathbb{R}^{n}$ of $\mathbb{R}^{n}$ that agrees with the cotangent fiber $T_{x}^{\ast }\mathbb{R}^{n}$ over a point $x\neq 0$ outside a compact set. Assume that $L$ is disjoint from the cotangent fiber at the origin. The projection of $L$ to the base extends to a map of the $n$-sphere $S^{n}$ into $\mathbb{R}^{n}\setminus \{0\}$. We show that this map is homotopically trivial, answering a question of Eliashberg. We give a number of generalizations of this result, including homotopical constraints on embedded Lagrangian disks in the complement of another Lagrangian submanifold, and on two-component links of immersed Lagrangian spheres with one double point in $T^{\ast }\mathbb{R}^{n}$, under suitable dimension and Maslov index hypotheses. The proofs combine techniques from Ekholm and Smith [Exact Lagrangian immersions with a single double point, J. Amer. Math. Soc. 29 (2016), 1–59] and Ekholm and Smith [Exact Lagrangian immersions with one double point revisited, Math. Ann. 358 (2014), 195–240] with symplectic field theory.Item Open Access Cutoff for conjugacy-invariant random walks on the permutation group(Springer Science and Business Media LLC, 2019) Berestycki, Nathanael; Sengul, BatiWe prove a conjecture raised by the work of Diaconis and Shahshahani (1981) about the mixing time of random walks on the permutation group induced by a given conjugacy class. To do this we exploit a connection with coalescence and fragmentation processes and control the Kantorovitch distance by using a variant of a coupling due to Oded Schramm. Recasting our proof in the language of Ricci curvature, our proof establishes the occurrence of a phase transition, which takes the following form in the case of random transpositions: at time $cn/2$, the curvature is asymptotically zero for $c\le 1$ and is strictly positive for $c>1$.Item Open Access Published version Peer-reviewed Intersection and mixing times for reversible chains(Institute of Mathematical Statistics, 2017) Peres, Y; Sauerwald, T; Sousi, P; Stauffer, A© 2017, University of Washington. All rights reserved. We consider two independent Markov chains on the same finite state space, and study their intersection time, which is the first time that the trajectories of the two chains intersect. We denote by tI the expectation of the intersection time, maximized over the starting states of the two chains. We show that, for any reversible and lazy chain, the total variation mixing time is O(tI). When the chain is reversible and transitive, we give an expression for tI using the eigenvalues of the transition matrix. In this case, we also show that tI is of order √nE[I], where I is the number of intersections of the trajectories of the two chains up to the uniform mixing time, and n is the number of states. For random walks on trees, we show that tI and the total variation mixing time are of the same order. Finally, for random walks on regular expanders, we show that tI is of order √n.Item Open Access Accepted version Peer-reviewed Chordal SLE6 explorations of a quantum disk(Institute of Mathematical Statistics, 2018) Gwynne, E; Miller, JWe consider a particular type of $\sqrt{8/3}$-Liouville quantum gravity surface called a doubly marked quantum disk (equivalently, a Brownian disk) decorated by an independent chordal SLE$_6$ curve $\eta$ between its marked boundary points. We obtain descriptions of the law of the quantum surfaces parameterized by the complementary connected components of $\eta([0,t])$ for each time $t \geq 0$ as well as the law of the left/right $\sqrt{8/3}$-quantum boundary length process for $\eta$.Item Open Access Accepted version Peer-reviewed CAPACITY OF THE RANGE OF RANDOM WALK ON Z(4)(American Mathematical Society, 2019-05) Asselah, Amine; Schapira, Bruno; Sousi, PerlaWe study the capacity of the range of a transient simple random walk on Z^d. Our main result is a central limit theorem for the capacity of the range for d \geq 6. We present a few open questions in lower dimensions.Item Open Access Accepted version Peer-reviewed Strong law of large numbers for the capacity of the Wiener sausage in dimension four(Springer Science and Business Media LLC, 2019-04) Asselah, Amine; Schapira, Bruno; Sousi, PerlaWe prove a strong law of large numbers for the Newtonian capacity of a Wiener sausage in the critical dimension four, where a logarithmic correction appears in the scaling. The main step of the proof is to obtain precise asymptotics for the expected value of the capacity. This re- quires a delicate analysis of intersection probabilities between two independent Wiener sausages.Item Open Access Accepted version Peer-reviewed Mirror Symmetry for Lattice Polarized del Pezzo Surfaces(International Press of Boston, 2018) Doran, Charles F; Thompson, Alan; Thompson, Alan [0000-0003-1400-0098]We describe a notion of lattice polarization for rational elliptic surfaces and weak del Pezzo surfaces, and describe the complex moduli of the former and the K\"{a}hler cone of the latter. We then propose a version of mirror symmetry relating these two objects, which should be thought of as a form of Fano-LG correspondence. Finally, we relate this notion to other forms of mirror symmetry, including Dolgachev-Nikulin-Pinkham mirror symmetry for lattice polarized K3 surfaces and the Gross-Siebert program.Item Open Access Published version Peer-reviewed Symplectomorphisms of exotic discs(Cellule MathDoc/CEDRAM, 2018-01-01) Casals, Roger; Keating, Ailsa; Smith, Ivan; Courte, Sylvain; Keating, Ailsa [0000-0002-1288-3117]We construct a symplectic structure on a disc that admits a compactly supported symplectomorphism which is not smoothly isotopic to the identity. The symplectic structure has an overtwisted concave end; the construction of the symplectomorphism is based on a unitary version of the Milnor--Munkres pairing. En route, we introduce a symplectic analogue of the Gromoll filtration. The Appendix by S. Courte shows that for our symplectic structure the map from compactly supported symplectic mapping classes to compactly supported smooth mapping classes is in fact surjective.Item Open Access Published version Peer-reviewed On the arithmetic of simple singularities of type E.(2018-01) Romano, Beth; Thorne, Jack; Romano, Beth [0000-0002-4480-636X]An ADE Dynkin diagram gives rise to a family of algebraic curves. In this paper, we use arithmetic invariant theory to study the integral points of the curves associated to the exceptional diagrams E 6 , E 7 , E 8 . These curves are non-hyperelliptic of genus 3 or 4. We prove that a positive proportion of each family consists of curves with integral points everywhere locally but no integral points globally.Item Open Access Accepted version Peer-reviewed Liouville quantum gravity as a metric space and a scaling limitMiller, JPOver the past few decades, two natural random surface models have emerged within physics and mathematics. The first is Liouville quantum gravity, which has its roots in string theory and conformal field theory from the 1980s and 1990s. The second is the Brownian map, which has its roots in planar map combinatorics from the 1960s together with recent scaling limit results. This article surveys a series of works with Sheffield in which it is shown that Liouville quantum gravity (LQG) with parameter $\gamma=\sqrt{8/3}$ is equivalent to the Brownian map. We also briefly describe a series of works with Gwynne which use the $\sqrt{8/3}$-LQG metric to prove the convergence of self-avoiding walks and percolation on random planar maps towards SLE$_{8/3}$ and SLE$_6$, respectively, on a Brownian surface.Item Open Access Published version Peer-reviewed On sensitivity of mixing times and cutoff(Institute of Mathematical Statistics, 2018) Hermon, Jonathan; Peres, Yuval; Hermon, Jonathan [0000-0002-2935-3999]A sequence of chains exhibits (total-variation) cutoff (resp., pre-cutoff) if for all $0<\epsilon< 1/2$, the ratio $t_{\mathrm{mix}}^{(n)}(\epsilon)/t_{\mathrm{mix}}^{(n)}(1-\epsilon)$ tends to 1 as $n \to \infty $ (resp., the $\limsup$ of this ratio is bounded uniformly in $\epsilon$), where $t_{\mathrm{mix}}^{(n)}(\epsilon)$ is the $\epsilon$-total-variation mixing-time of the $n$th chain in the sequence. We construct a sequence of bounded degree graphs $G_n$, such that the lazy simple random walks (LSRW) on $G_n$ satisfy the "product condition" $\mathrm{gap}(G_n) t_{\mathrm{mix}}^{(n)}(\epsilon) \to \infty $ as $n \to \infty$, where $\mathrm{gap}(G_n)$ is the spectral gap of the LSRW on $G_n$ (a known necessary condition for pre-cutoff that is often sufficient for cutoff), yet this sequence does not exhibit pre-cutoff. Recently, Chen and Saloff-Coste showed that total-variation cutoff is equivalent for the sequences of continuous-time and lazy versions of some given sequence of chains. Surprisingly, we show that this is false when considering separation cutoff. We also construct a sequence of bounded degree graphs $G_n=(V_{n},E_{n})$ that does not exhibit cutoff, for which a certain bounded perturbation of the edge weights leads to cutoff and increases the order of the mixing-time by an optimal factor of $\Theta (\log |V_n|)$. Similarly, we also show that "lumping" states together may increase the order of the mixing-time by an optimal factor of $\Theta (\log |V_n|)$. This gives a negative answer to a question asked by Aldous and Fill.